Topology

Topologie

6 EC

Semester 2, period 4, 5

5122TOPO6Y

Owner Bachelor Wiskunde
Coordinator M. Hlushchanka
Part of Exchange Programme Faculty of Science, specialisation BSc Mathematics, year 1Minor Mathematical Themes, year 1Bachelor Mathematics, year 2Double bachelor Mathematics and Physics, year 2Double bachelor Mathematics and Computer Science, year 2Bachelor Bèta-gamma, major Wiskunde, year 3Bachelor Econometrics and Data Science, specialisation Data Science, year 3Bachelor Econometrics and Data Science, specialisation Econometrics, year 3
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Course manual 2025/2026

Course content

In this course, the fundamentals of topology are treated. Various concepts that play a role in analysis are reflected in this course in an abstract form. The course lays a foundation for the further study of geometry, algebraic topology, and differential topology. Topics covered include topological spaces, continuous maps and homeomorphisms, connectedness, compactness, and quotient spaces. The fundamental group is discussed in detail. In particular, we compute the fundamental group of a number of well-known spaces and study the relationship between fundamental groups and covering spaces.

Study materials

Literature

  • J.R. Munkres, 'Topology', Prentice Hall Inc., 2nd edition, 2000, ISBN 0-13-181629-2.

Objectives

  • Algemene Topologie: de student kent definities, voorbeelden en basiseigenschappen van de volgende concepten: topologie, open, gesloten, sluiting, basis, Hausdorff (H), compact (C), samenhangend (S), padsamenhangend, continue afbeeldingen, homeomorfisme, deelruimte-topologie, product-topologie, quotiënt-topologie.
  • Algebraische Topologie: de student kent definities, voorbeelden en basiseigenschappen van de volgende concepten: path, homotopie, retractie, overdekking, liften van afbeeldingen, de fundamentaalgroep, enkelvoudig samenhangend.
  • Stellingen: Studenten kunnen de volgende belangrijke stellingen formuleren, bewijzen en toepassen: plaklemma, gedrag van H/S/C onder producten en continue afbeeldingen, samenhangende en compacte deelverzamelingen in R, karakterisatie van de quotiëntafbeelding, classificatie van oppervlakken, fundamentaalgroep van de cirkel en enkele oppervlakken. Brouwer’s dekpuntstelling, de hoofdstelling van de algebra.
  • De student kan verschillende topologische eigenschappen van topologische ruimten nagaan en bewijzen.
  • De student begrijpt het belang van topologische technieken bij het bestuderen van problemen en de invloed van topologische eigenschappen op het gedrag en het bestaan van oplossingen.

Teaching methods

  • Self-study
  • Lecture

Additional materials provided on Canvas

Learning activities

Activiteit

Aantal uur

Hoorcollege

28

Tentamen

3

Tussentoets

2

Werkcollege

28

Zelfstudie

104

Attendance

Attendance requirements for the program (OER - Part B):

  • Active participation is expected from every student in the course component for which the student is enrolled.
  • In addition to the general requirement that the student actively participates in the education, the additional requirements per component are described in the study guide. It also specifies which parts of the component have a mandatory attendance requirement.
  • If a student is unable to attend a mandatory part of the program due to personal circumstances, the student must report this in writing as soon as possible to the relevant lecturer and the study advisor.
  • It is not permitted to miss mandatory parts of a component if there are no personal circumstances.
  • In cases of qualitatively or quantitatively insufficient participation, the examiner may exclude the student from further participation in the component or part of it. Conditions for sufficient participation are determined in advance in the study guide and on Canvas.

Assessment

Item and weight Details

Final grade

1 (100%)

Deeltoets

The midterm test covers point-set topology and counts for 25%.

The Final Exam covers both point-set topology and algebraic topology and counts for 60%.

The homework counts for 15%.

 

The resit exam counts 100% towards the final grade (i.e., the midterm and homework will not be counted).

Fraud and plagiarism

The 'Regulations governing fraud and plagiarism for UvA students' applies to this course. This will be monitored carefully. Upon suspicion of fraud or plagiarism the Examinations Board of the programme will be informed. For the 'Regulations governing fraud and plagiarism for UvA students' see: www.student.uva.nl

Course structure

Here is a tentative overview of the course with the corresponding sections from the text by Munkres that will be covered.

 

Week 1: Topological spaces, examples of topological spaces, basis for a topology, relationship with metric spaces (sections 12, 13, 20).

Week 2: Product topology, subspace topology, interior and closure (sections 15,16,17).

Week 3: Hausdorff spaces, continuous functions (sections 17 and 18).

Week 4: Connected spaces (sections 23, 24, 25)

Week 5: Compactness (sections 26, 27).

Week 6 Quotient topology (section 22).

Week 7: Review and (possibly) application on surfaces: cut and paste (section 80)

 

Week 8: Midterm test

 

Week 9: Homotopy of continuous images and of paths (section 51).

Week 10: The Fundamental Group and Cover Spaces (sections 52 and 53).

Week 11: The fundamental group of the circle (section 54).

Week 12: Retractions (sections 55.1-6, 58.1-3)

Week 13: The fundamental group of some surfaces (section 59.60)

Week 14: The universal covering of a surface and the fundamental group

Week 15: Review

Honours information

There is an honors extension worth 3 EC for the Topology course.
In the Honors extension, students will apply surface classification themselves.

Contact information

Coordinator

  • M. Hlushchanka

Docenten

  • Mikhail Hlushchanka

Staff

  • Arix Eggink
  • Koen Hoeberechts
  • Annika Holtrup
  • Mollie Jagoe Brown